Proof. Suppose first \(\varphi :=\mathcal {F}\psi \in \mathcal {C}^\infty _c(\mathbb {R}^3,4)\) . Let \(K:=\operatorname{supp}(\varphi )\) . Set \(\gamma :=\max \lbrace \frac{|p|}{\epsilon (p)}:p\in K\rbrace \) . Clearly \(0<\gamma <1\) . For the estimation of the integral in Eq. (REF ) consider \(\phi ^\eta (p):= (|x|+|t|)^{-1}\big (px-t\eta \epsilon (p)\big )\) . Then \(\nabla \phi ^\eta (p)= (|x|+|t|)^{-1}\big (x-\frac{t\eta }{\epsilon (p)}p\big )\)  and \(|\nabla \phi ^\eta (p)|\ge (|x|+|t|)^{-1}\big (|x|-|t|\frac{|p|}{\epsilon (p)}\big )\ge \frac{|x|-\gamma |t|}{|x|+|t|}\)  for \(p\in K\) . Now suppose \(|x|\ge |t|\) . Then \(|\nabla \phi ^\eta (p)|\ge \frac{|x|-\gamma |x|}{|x|+|x|}=\frac{1-\gamma }{2}>0\) . This implies (cf. [1]}) for \(\eta \in \lbrace +,-\rbrace \) , \(l=1,\dots ,4\) , and for every \(N>0\)  that there is a finite constant \(A_N\)  with
\(\big |\big (\psi ^\eta _t(x)\big )_l\big |\le A_N(1+|x|+|t|)^{-N}\textrm { \;if\; } |x|\ge |t|\) 
